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On Some Non-Linear Problems

Published online by Cambridge University Press:  20 November 2018

K. Srinivasacharyulu
K. Srinivasacharyulu*
Affiliation:
Université de Montréal, Montréal
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Non-linear problems have been studied by Krasnoselski, Browder, and others; in fact Browder and independently Kirk (cf., 1; 5) have proved the following remarkable theorem: let X be a uniformly convex Banach space, U a non-expansive mapping of a bounded closed convex subset C of X into C, i.e., ||Ux — Uy|| ⩽ ||x — y|| for x, y ∊ C; then U has a fixed point in C. The aim of this paper is to give some existence theorems for non-linear functional equations in uniformly convex Banach spaces. Similar results may be found in (3 ; 6).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 394 - 397
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

*

Supported by the Canadian Mathematical Congress while at the Summer Research Institute.

References

1. Browder, F. E., Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 64 (1965), 1041–1044.Google Scholar
2. Edelstein, M., On nonexpansive mappings of Banach spaces, Proc. Cambridge Philos. Soc, 60 (1964), 439–447.Google Scholar
3. Granas, A., The theory of compact vector fields and some of its applications to topology of functional spaces I, Vol. xxx, Rozprawy Mathmatyczne, Polska Akademia Nauk (1962), pp. 63–64.Google Scholar
4. James, R. C., Weak compactness and separation. Can. J. Math., 16 (1964), 204–206.Google Scholar
5. Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006.Google Scholar
6. Krasnoselskii, M. A., Topological methods in the theory of non-linear integral equations (Pergamon Press, 1964).Google Scholar